27/05/2019
IPCC (2012)
The marginal distribution of temperature extremes is changing -
Is the dependence also changing?
If so by how much?
What does this mean for events like heatwaves?
Marginals
Dependence
Observations
Climate Models
Let \(M\) hottest 3-day average temperature in a year: \[
\mathbb{P}(M < z) = \exp \left\{ - \left[ 1 + \xi \left(\dfrac{z-\mu}{\sigma}\right) \right]_+^{-1 / \xi} \right\},\] where \([v]_+ = \max \left\lbrace 0,v \right\rbrace\), \(\mu \in \mathbb{R}\), \(\sigma \in \mathbb{R}^+\) and \(\xi \in \mathbb{R}\).
(Fisher and Tippett 1928, Gnendenko 1943)
Address the trend in changing temperature
First example: Model changing in location linearly
Later: (Lazy) Assume stationarity in a given period
For dependence we want common marginals
Wlog we can remove the trend and transform the marignal distributions
\(\mathbb{P}(M < z) = \exp(-1/z)\)
Now dependence!
Assume the distribution of \((M_{i}, M_{j})\) can be approximated with a bivariate extreme value theory distribution, then
\[\mathbb{P}\left( M_{i} \leq z_i,\, M_{j} \leq z_j \right) = \exp\left\{ - V_{ij} \left( \dfrac{-1}{\log F_i(z_i)}, \, \dfrac{-1}{\log F_j(z_j)}\right) \right\}\] where the exponent measure \(V_{ij}(a,b)\) is given by \[V_{ij}(a, b) = 2 \int_0^1 \max \left(\dfrac{w}{a}, \dfrac{1-w}{b} \right) \text{d}H_{ij}(w)\] and \(H\) is any distribution function on \([0,1]\) with expectation equal to 0.5.
Lots of parameteric options to specify form of \(V_{ij}(a, b)\), but
Want to avoid assuming that a parametric form that is constant in time
Want a solution that will work for lots of different pairs
so …
Nonparametric
\[V_{ij}(z) = \left(\frac{1}{z_i} + \frac{1}{z_j}\right)A\left(\frac{z_i}{|z_i + z_j|} + \frac{z_j}{|z_i + z_j|}\right)\]
C1: \(A(t)\) is continuous and convex for \(t \in [0,1]\)
C2: Bounds on the function of \(\max \{(1 - t), t \} \leq A(t) \leq 1\)
C3: Boundary conditions of \(A(0) = A(1) = 1\)
\[\mathbb{P}\left(M_{i} \leq z, M_{j} \leq z\right) = \left[\mathbb{P}(M_{i} \leq z)\mathbb{P}(M_{j} \leq z) \right]^{A\left(1/2\right)}.\]
Model information
22 Global Climate Model - Regional Cliamte Model combinations from EURO-CORDEX
0.11 degree simulation
rotated-pole grid
1950-2005 with historical forcing
2006-2100 with RCP 8.5
Consider Annual Maxima
Pairs relative to De Bilt, NL
Years modelled: 1950 - 2000
Strong initial evidence to suggest:
Examine assumptions on the marginal distriubtion
Fit dependence as a function of time
Think about how to translate insights from extreme value statistics into insights about changing heat extremes
That includes 'event' definition
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